The quasi-optimality criterion in the linear functional strategy

TL;DR

This paper introduces a heuristic quasi-optimality principle for regularization parameter selection in inverse problems, providing convergence analysis, noise condition insights, and an adaptive aggregation method with improved numerical performance.

Contribution

It proposes a new heuristic rule for the linear functional strategy, analyzes its convergence and noise conditions, and introduces an adaptive aggregation method for better parameter choice.

Findings

Convergence rates are established under the heuristic rules considering solution and functional smoothness.

Noise conditions are satisfied almost surely in mildly-ill-posed problems with Gaussian noise.

The adaptive aggregation method outperforms standard heuristic rules in numerical experiments.

Abstract

The linear functional strategy for the regularization of inverse problems is considered. For selecting the regularization parameter therein, we propose the heuristic quasi-optimality principle and some modifications including the smoothness of the linear functionals. We prove convergence rates for the linear functional strategy with these heuristic rules taking into account the smoothness of the solution and the functionals and imposing a structural condition on the noise. Furthermore, we study these noise conditions in both a deterministic and stochastic setup and verify that for mildly-ill-posed problems and Gaussian noise, these conditions are satisfied almost surely, where on the contrary, in the severely-ill-posed case and in a similar setup, the corresponding noise condition fails to hold. Moreover, we propose an aggregation method for adaptively optimizing the parameter choice rule by making use of improved rates for linear functionals. Numerical results indicate that this method yields better results than the standard heuristic rule.